Is 2sin(30)cos(30)=sin(60) ?

The answer to whether 2sin(30)cos(30)=sin(60) is True

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prove 2sin(30)cos(30)=sin(60)

prove

2 sin (3 0^{\circ}) cos (3 0^{\circ}) = sin (6 0^{\circ})

True

Solution steps

2 sin (3 0^{\circ}) cos (3 0^{\circ}) = sin (6 0^{\circ})

Manipulating left side

2 sin (3 0^{\circ}) cos (3 0^{\circ})

Simplify

2 cos (3 0^{\circ}) sin (3 0^{\circ}) : \frac{3}{2}

2 cos (3 0^{\circ}) sin (3 0^{\circ})

Simplify

cos (3 0^{\circ}) : \frac{3}{2}

cos (3 0^{\circ})

Use the following trivial identity:

cos (3 0^{\circ}) = \frac{3}{2}

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

= 2 \cdot \frac{3}{2} sin (3 0^{\circ})

Simplify

sin (3 0^{\circ}) : \frac{1}{2}

sin (3 0^{\circ})

Use the following trivial identity:

sin (3 0^{\circ}) = \frac{1}{2}

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

= 2 \cdot \frac{1}{2} \cdot \frac{3}{2}

Simplify

2 \cdot \frac{3}{2} \cdot \frac{1}{2}

Multiply fractions:

a \cdot \frac{b}{c} \cdot \frac{d}{e} = \frac{a \cdot b \cdot d}{c \cdot e}

= \frac{3 \cdot 2}{2 \cdot 2}

Cancel the common factor:

2

= \frac{3}{2}

= \frac{3}{2}

= \frac{3}{2}

Manipulating right side

sin (6 0^{\circ})

Simplify

= \frac{3}{2}

We showed that the two sides could take the same form

\Rightarrow True

Is 2sin(30)cos(30)=sin(60) ?
The answer to whether 2sin(30)cos(30)=sin(60) is True